ﻻ يوجد ملخص باللغة العربية
We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.
Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most
We give a bound on the spectral radius of a graph implying a quantitative version of the Erdos-Stone theorem.
Given $r$-uniform hypergraphs $G$ and $H$ the Turan number $rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the ErdH{o}s-Renyi random $r$-uniform hypergra
We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.
The Turan number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P